What is the integral of (ln(x))/(2sqrt(x)) ?

The integral of (ln(x))/(2sqrt(x)) is sqrt(x)(ln(x)-2)+C

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integral of (ln(x))/(2sqrt(x))

\int \frac{ln ( x )}{2 x} d x

x (ln (x) - 2) + C

Solution steps

\int \frac{ln ( x )}{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{ln ( x )}{x} d x

Apply Integration By Parts

: 2 x ln (x) - \int \frac{2}{x} d x

= \frac{1}{2} (2 x ln (x) - \int \frac{2}{x} d x)

\int \frac{2}{x} d x = 4 x

= \frac{1}{2} (2 x ln (x) - 4 x)

Simplify

\frac{1}{2} (2 x ln (x) - 4 x) : x (ln (x) - 2)

= x (ln (x) - 2)

Add a constant to the solution

= x (ln (x) - 2) + C

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What is the integral of (ln(x))/(2sqrt(x)) ?
The integral of (ln(x))/(2sqrt(x)) is sqrt(x)(ln(x)-2)+C